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Polya's Urn

Q: An urn has \(r\) red balls and \(b\) blue balls. Someone draws a ball at random, its colour observed and put back into the urn. You do not know what was observed. However that person puts back \(x\) balls of the same colour back into the urn. Now, you draw a second ball from this urn. What is the probability that it is red?

A: The framing of this puzzle follows directly from the "Polya's Urn" process. It presents yet another surprising result from Bayesian reasoning. Intuitively, it appears that the act of putting in new balls of the same colour would tamper with the probability of drawing a red ball for the second draw. But does it? Lets take a look.

The probability that a red ball is drawn from the urn in the first draw is \(\frac{r}{r+b}\) and for a blue ball would be \(\frac{b}{r+b}\). The second draw, if it is a red ball, could be a consequence of either a red ball being drawn the first time or a blue ball.

For the second draw, the probability that a red ball is drawn if a red ball is drawn the first time, would be \(\frac{r+x}{r + b + x}\). The probability that a red ball is drawn if a blue ball is drawn the first time, would be \(\frac{b}{r+b+x}\). This layout is shown in the figure below.

The probability that a red ball is drawn on the second draw is
$$
P(\text{Red: Draw=2})=\frac{r + x}{r + b + x}\times\frac{r}{r+b} + \frac{r}{r+b+x}\times\frac{b}{r+b}
$$
The above simplifies as
$$
\frac{(r+b+x)r}{(r+b+x)(r+b)} = \frac{r}{r+b}
$$
Note, the probability remains exactly the same!
As with most Bayesian stuff, the result eventually becomes intuitive when we spend more time thinking of the problem. The act of adding \(x\) balls based on the outcome of the first draw, is really meaningless!

Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

Linear Algebra (Dover Books on Mathematics)
An excellent book to own if you are looking to get into, or want to understand linear algebra. Please keep in mind that you need to have some basic mathematical background before you can use this book.

Linear Algebra Done Right (Undergraduate Texts in Mathematics)
A great book that exposes the method of proof as it used in Linear Algebra. This book is not for the beginner though. You do need some prior knowledge of the basics at least. It would be a good add-on to an existing course you are doing in Linear Algebra.

Follow @ProbabilityPuzIf you are looking to learn time series analysis, the following are some of the best books in time series analysis.

Introductory Time Series with R (Use R!)
This is good book to get one started on time series. A nice aspect of this book is that it has examples in R and some of the data is part of standard R packages which makes good introductory material for learning the R language too. That said this is not exactly a graduate level book, and some of the data links in the book may not be valid.

Econometrics
A great book if you are in an economics stream or want to get into it. The nice thing in the book is it tries to bring out a oneness in all the methods used. Econ majors need to be up-to speed on the grounding mathematics for time series analysis to use this book. Outside of those prerequisites, this is one of the best books on econometrics and time series analysis.